How can I extrude a egde into the middle of another egde?
Blender has a built in tool to join faraway verts such as these, and to automatically create all the connecting edges. The tool is called "Connect Vertex Path", and is assigned the shortcut key J.

I find it to be a pretty essential modeling tool I personally discovered rather late.

All you need to do to use it, is to select two verts, then press J: